3.2097 \(\int \frac{x}{\left (a+\frac{b}{x^4}\right )^{5/2}} \, dx\)

Optimal. Leaf size=64 \[ \frac{4 x^2 \sqrt{a+\frac{b}{x^4}}}{3 a^3}-\frac{2 x^2}{3 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^2}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

[Out]

-x^2/(6*a*(a + b/x^4)^(3/2)) - (2*x^2)/(3*a^2*Sqrt[a + b/x^4]) + (4*Sqrt[a + b/x
^4]*x^2)/(3*a^3)

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Rubi [A]  time = 0.0800754, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4 x^2 \sqrt{a+\frac{b}{x^4}}}{3 a^3}-\frac{2 x^2}{3 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^2}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^4)^(5/2),x]

[Out]

-x^2/(6*a*(a + b/x^4)^(3/2)) - (2*x^2)/(3*a^2*Sqrt[a + b/x^4]) + (4*Sqrt[a + b/x
^4]*x^2)/(3*a^3)

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Rubi in Sympy [A]  time = 6.73961, size = 56, normalized size = 0.88 \[ - \frac{x^{2}}{6 a \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}} - \frac{2 x^{2}}{3 a^{2} \sqrt{a + \frac{b}{x^{4}}}} + \frac{4 x^{2} \sqrt{a + \frac{b}{x^{4}}}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b/x**4)**(5/2),x)

[Out]

-x**2/(6*a*(a + b/x**4)**(3/2)) - 2*x**2/(3*a**2*sqrt(a + b/x**4)) + 4*x**2*sqrt
(a + b/x**4)/(3*a**3)

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Mathematica [A]  time = 0.0357642, size = 51, normalized size = 0.8 \[ \frac{3 a^2 x^8+12 a b x^4+8 b^2}{6 a^3 x^2 \sqrt{a+\frac{b}{x^4}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x^4)^(5/2),x]

[Out]

(8*b^2 + 12*a*b*x^4 + 3*a^2*x^8)/(6*a^3*Sqrt[a + b/x^4]*x^2*(b + a*x^4))

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Maple [A]  time = 0.012, size = 50, normalized size = 0.8 \[{\frac{ \left ( a{x}^{4}+b \right ) \left ( 3\,{x}^{8}{a}^{2}+12\,ab{x}^{4}+8\,{b}^{2} \right ) }{6\,{a}^{3}{x}^{10}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^4)^(5/2),x)

[Out]

1/6*(a*x^4+b)*(3*a^2*x^8+12*a*b*x^4+8*b^2)/a^3/x^10/((a*x^4+b)/x^4)^(5/2)

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Maxima [A]  time = 1.44388, size = 73, normalized size = 1.14 \[ \frac{\sqrt{a + \frac{b}{x^{4}}} x^{2}}{2 \, a^{3}} + \frac{6 \,{\left (a + \frac{b}{x^{4}}\right )} b x^{4} - b^{2}}{6 \,{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} a^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a + b/x^4)^(5/2),x, algorithm="maxima")

[Out]

1/2*sqrt(a + b/x^4)*x^2/a^3 + 1/6*(6*(a + b/x^4)*b*x^4 - b^2)/((a + b/x^4)^(3/2)
*a^3*x^6)

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Fricas [A]  time = 0.241041, size = 88, normalized size = 1.38 \[ \frac{{\left (3 \, a^{2} x^{10} + 12 \, a b x^{6} + 8 \, b^{2} x^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{6 \,{\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a + b/x^4)^(5/2),x, algorithm="fricas")

[Out]

1/6*(3*a^2*x^10 + 12*a*b*x^6 + 8*b^2*x^2)*sqrt((a*x^4 + b)/x^4)/(a^5*x^8 + 2*a^4
*b*x^4 + a^3*b^2)

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Sympy [A]  time = 7.8932, size = 163, normalized size = 2.55 \[ \frac{3 a^{2} b^{\frac{9}{2}} x^{8} \sqrt{\frac{a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} + \frac{12 a b^{\frac{11}{2}} x^{4} \sqrt{\frac{a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} + \frac{8 b^{\frac{13}{2}} \sqrt{\frac{a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x**4)**(5/2),x)

[Out]

3*a**2*b**(9/2)*x**8*sqrt(a*x**4/b + 1)/(6*a**5*b**4*x**8 + 12*a**4*b**5*x**4 +
6*a**3*b**6) + 12*a*b**(11/2)*x**4*sqrt(a*x**4/b + 1)/(6*a**5*b**4*x**8 + 12*a**
4*b**5*x**4 + 6*a**3*b**6) + 8*b**(13/2)*sqrt(a*x**4/b + 1)/(6*a**5*b**4*x**8 +
12*a**4*b**5*x**4 + 6*a**3*b**6)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a + b/x^4)^(5/2),x, algorithm="giac")

[Out]

integrate(x/(a + b/x^4)^(5/2), x)